Question

Graph the piecewise-defined function and use your graph to find the values of the limits, if they exist.

$f\left(x\right)=\left\{\begin{array}{l}\text{2}if\text{x<0}\\ \text{x+1}ifx\ge 0\end{array}\right.$

$\left(a\right)\underset{x\to 0^{-}}{\lim }f\left(x\right)\left(b\right)\underset{x\to 0^{+}}{\lim }f\left(x\right)\left(c\right)\underset{x\to 0}{\lim }f\left(x\right)$

Short answer

Limit doesn’t exist.

Step-by-step solution

Step 1. Given information.

The limit of functions here given are,

$f\left(x\right)=\left\{\begin{array}{l}\text{2}if\text{x<0}\\ \text{x+1}ifx\ge 0\end{array}\right.$

Step 2. Graph.

The graph of is shown in the figure.

Step 3. Details and conclusion.

From the graph, we see that the values of$f\left(x\right)$approach$2$as$x$approaches $0$from the left, but they approach$1$as$x$approaches$0$from the right.

Thus, the left- and right-hand limits are not equal.

So we have,

$\underset{x\to 0^{-}}{\lim }f\left(x\right)=2$and $\underset{x\to 0^{+}}{\lim }f\left(x\right)=1$

Therefore,$\underset{x\to 0}{\lim }f\left(x\right)$doesn’t exist.